SPACELIKE HYPERSURFACES IN RIEMANNIAN OR LORENTZIAN SPACE FORMS SATISFYING Lkx = Ax + b

We study connected orientable spacelike hypersurfaces x : M^n -- M n+1 q (c), isometrically immersed into the Riemannian or Lorentzian space forms of curvature c = -1, 0, 1, and index q = 0, 1, satisfying the condition Lkx = Ax + b, where Lk is the linearized operator of the (k+1)th mean curvature Hk+1 of the hypersurface for a fixed integer 0 ≤ k n, A is a constant matrix and b is a constant vector. We show that the only hypersurfaces satisfying that condition are hypersurfaces with zero Hk+1 and constant Hk (when c ≠ 0), open pieces of totally umbilic hypersurfaces and open pieces of the standard Riemannian product of two totally umbilic hypersurfaces.